Vortex Dynamics and Dissipation under High-Amplitude Microwave Drive

In this paper, we describe the vortex dynamics under a high-amplitude microwave drive and its effect on the surface resistance of superconductors. The vortex surface resistance is calculated with a Monte Carlo approach, where the vortex equation of motion is solved for a collection of vortex flux lines, each oscillating within a random pinning landscape. This approach is capable of providing a detailed description of the microscopic vortex dynamics and in turn important insights into the microwave-field-amplitude dependence of the vortex surface resistance. The numerical simulations are compared against experimental data of vortex surface resistance at high microwave amplitude measured by means of bulk niobium superconducting radio-frequency cavities operating at 1.3 GHz. The good qualitative agreement of the simulations and experiments suggests that the nonlinear dependence of the trapped-flux surface resistance with the microwave field amplitude is generated by progressive microwave depinning and vortex jumps.

Figure 1

(a) The experimental data of the trapped-flux surface resistance up to high values of the peak magnetic field. (b) The normalized $R_{\mathrm{fl}}$ over the trapped magnetic field.

Figure 2

(a) A sketch of the cavity section, indicating the locations of the thermometers. (b) The ratio between the local magnetic field at the surface over the peak magnetic field at each thermometer location. (c) The Tmap, showing the vortex dissipation collected at $B_p=166$ mT.

Figure 3

An example of vortex-dynamics simulation. (a) The phase portrait of the vortex tip. The arrows indicate the direction of the vortex motion. The transition from the initial condition to the steady state is shown in light blue. (b) The time evolution of the vortex displacement with respect to its initial position is shown by the dashed line. The solid lines show the vortex motion in the steady state, while the dotted lines show the time evolution from initial condition to steady state. The pinning landscape in the background is calculated by means of Eq. (7), stronger pinning areas being shown with a darker contrast.

Figure 4

An example of a vortex-surface-resistance simulation calculated for $l=300$ nm and $f_{p,\mathrm{ave}}=200\mu \mathrm{N}/$m as a function of the peak field. (a) The average single-vortex power dissipation ($⟨P^{(1)}⟩$), calculated as a function of the peak magnetic field. (b),(c) The average phase portraits calculated over 100 vortex simulations at 0.001 mT and 200 mT, respectively. The parameter $u_0$ represents the $x$ offset of the oscillation center. The thin colored solid line represents the individual vortex phase portrait, the thick solid black line their average, and the white dash-dotted line the phase portrait in the absence of pinning. (d) The simulated $R_{\mathrm{fl}}/B_0$ curve is shown with its abscissa in a semilogarithmic scale. In the inset, the same curve is shown in the range $1\mathrm{n}\mathrm{\Omega }\mathrm{/mG}\le R_{\mathrm{fl}}/B_0\le 1.8\mathrm{n}\mathrm{\Omega }\mathrm{/mG}$, with its abscissa in a linear scale. In (a) and (d), the dotted line above 200 mT shows how the trend evolves to saturation above the expected maximum $B_p$ value that $\mathrm{Nb}$ can sustain, since for the selected pinning landscape, saturation is not achieved below 200 mT.

Figure 5

An example of single-vortex dynamics as a function of the microwave field. (a) The single-vortex power dissipation as a function of the peak magnetic field. (b)–(d) Phase portraits and vortex displacements as a function of the time calculated at the field values $B_p=63$ mT, $75$ mT, and $170$ mT. In (a), the respective values of the single-vortex dissipated power are highlighted by red dots.

Figure 6

A comparison of the simulated normalized vortex surface resistance with the experimental data. (a) Simulations for different values of the average pinning force at a fixed mean free path and a fixed factor $b$. (b) Simulations as a function of the mean-free-path value for a fixed average pinning force and factor $b$. (c) Simulations as a function of the factor $b$ for a fixed average pinning force and mean free path.

Figure 7

The normalized experimental data $\mathrm{\Delta }{T}_{\mathrm{out}}/B_p^2$ and the normalized simulated $R_{\mathrm{fl}}$ plotted against the angle between the microwave currents and the magnetic flux directions.